Narrow Results

On a Riemannian almost product manifold (M, P, g), we consider a linear connection preserving the almost product structure P and the Riemannian metric g and having a totally skew-symmetric torsion. We determine the class of the manifolds (M, P, g) admitting such a connection and prove that this connection is unique in terms of the covariant derivative of P with respect to the Levi-Civita connection. We find a necessary and sufficient condition the curvature tensor of the considered connection to have similar properties like the ones of the Kähler tensor in Hermitian geometry. We pay attention to the case when the torsion of the connection is parallel. We consider this connection on a Riemannian almost product manifold (G, P, g) constructed by a Lie group G.

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

In this paper, we construct metallic Kähler and nearly metallic Kähler structures on Riemannian manifolds. For such manifolds with these structures, we study curvature properties. Also, we describe linear connections on the manifold which preserve the associated fundamental $2$-form and satisfy some additional conditions and present some results concerning them.

In this paper, we give an analogue of the Hermitian structure in the almost complex case, on an $n(k+1)$-dimensional manifold endowed with an almost $k$-complex metric. Also, we study the compatibility between Riemannian metric and polarized $k$-symplectic structure. Also, we study some properties of an almost $k$-complex structure. Moreover, we give an equivalence between almost $k$-complex structures, $k$-almost tangent structures and $k$-almost cotangent structures.

The aim of this paper is to define and to construct some ∞-Lie functors from two categories of anchored modules which admit linear connections to the corresponding categories of Lie pseudoalgebras; the isomorphism class of the Lie pseudoalgebra does not depend on the linear connection or on the corresponding bracket. In the particular case of a module, the ∞-Lie algebra is isomorphic with the free Lie algebra of the module in Bourbaki's sense.

We show how the Chern character of the tangent bundle of a smooth manifold may be extracted from the geodesic distance function by means of cyclic homology. Such considerations lead us to define in Sect. 5 the notion of linear direct connection in vector bundles, notion which generalizes the notion of linear connection.

The basic difference between a linear direct connection and a linear connection consists of the fact that while a linear connection provides a transport of fibers along curves, a linear direct connection provides a direct transport of fibres from point to point.

For this reason, linear direct connections could be defined in contexts where differentiability is not available.

We show next that the algebraic procedure for constructing the Chern character, discussed in Sect. 4 applies also in the case of linear direct connections.

This paper provides a geometric interpretation of the non commutative Chern character due to A. Connes [1,2]. We show that this interpretation goes along the same lines as those presented by N. Teleman [11] and C. Teleman [10].

The arguments discussed here may be extended to the language of groupoids. In a subsequent paper we are going to improve some of the considerations presented here and extend their field of application to more singular situations.

Remark 0.1. The notion of linear direct connection, introduced in this paper, replaces the notion of linear quasi connection, introduced by the author in [12].

Our aim is to examine almost geodesic mappings of affine manifolds and give necessary and sufficient conditions for existence of the so-called mappings (canonical almost geodesic mappings of type π according to Sinyukov) of a manifold endowed with a linear connection onto pseudo-Riemannian manifolds. The conditions take the form of a closed system of PDE's of first order of Cauchy type. Our result is a generalization of some previous theorems of N.S. Sinyukov.